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<b>let </b><font color="Maroon">G1</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">G2</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">C1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">C2</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">G2</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="circtrm1.html#R4" target="_self">form_embedding_of</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="circtrm1.html#R3" target="_self">preserves_inputs_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 ;<br/>

<b>let </b><font color="Maroon">s2</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C2</font>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C1</font>;<br/>



<b>assume </b><a NAME="E3:74"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>;<br/>
<a NAME="E4:74"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T11">FACIRC_1:11</a></i>;<br/>
<a NAME="E5:74"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="circtrm1.html#T7" target="_self">Th7</a>, <a class="ref" href="facirc_1.html#T11">FACIRC_1:11</a></i>;<br/>
<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:74_1"/><b>then </b>
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="circtrm1.html#T6" target="_self">Th6</a>, , <a class="ref" href="circtrm1.html#T5" target="_self">Th5</a></i>;<br/><b>then </b> <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
;<br/><b>hence </b><a NAME="E4:74_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E7:74"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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